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Sma2110:Real Analysis Question Paper
Sma2110:Real Analysis
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2011
QUESTION ONE (30 MARKS)
a) Let A and B be sets. Define the following terms as used in sets i. ?????? (2 Marks) ii. ?? n ?? (2 Marks) iii. ?? ? ?? (2 Marks) iv. ?? ?? (2 Marks) b) Prove that the square of an odd number is odd. (3 Marks) c) Let ?? ? R2 and define ??:R2 ? R2 by ?? ??1,??1 ??2,??2 = ??2 - ??1 2 + ??2 - ??1 2 for all points ??1,??1 and ??2,??2 in R2. Show that d is a metric on R2. (4 Marks) d) Let ???? = 1 2 , 1 2 ,- 1 2 , 1 3 , 2 3 ,- 2 3 , 1 4 , 3 4 ,- 3 4 , 1 5 , 4 5 ,- 4 5 ,…. . find the set S of sub sequential limits, limit superior and limit inferior.(4 Marks) e) Show that ?? ?? = 3?? - 5 is uniformly continuous on -1,1 . (3 Marks)
f) Let ???? =
?? 2??
,???? ?? ???? ??????
1 2??
,???? ?? ???? ????????
Determine whether ???? converges. (4 Marks) g) Determine the interior of the sets: i. [7, 10] (2 Marks) ii. ?? = ?? 2,?? ? N . (2 Marks)
2
QUESTION TWO (20 MARKS)
a) i) Define an open set. (3 Marks) ii) Prove that the intersection of finite collection of open sets is open. (5 Marks) b) Use the set ???? = 2 - 1 ?? ,3 + 1 ?? :??? ? N to show that the intersection of an arbitrary collection of open sets is not necessarily open. (4 Marks) c) i) Define a limit point of a subset of R. (3 Marks) ii) Prove that finite sets have no limit points. (5 Marks)
QUESTION THREE (20 MARKS)
a) i) Define the term convergence of sequences. (3 Marks)
ii) Show from first principles that 3 1 23 lim n n n
. (4 Marks)
b) Prove that a sequence ???? in R has a unique limit. (5 Marks) c) Use the sequence ???? = - ?? 1 + 1 ?? ??? ? N, to show that boundedness does not imply convergence. (4 Marks)
d) Compute n n n n 2 5 34 lim 2 2
(4 Marks)
QUESTION FOUR (20 MARKS)
a) Define the following terms as used in series: i. Absolute convergence. (2 Marks) ii. Conditional convergence. (2 Marks) b) Determine whether (-1)??+1 ?? 8 ??=1 is conditionally convergent. (4 Marks) c) i) Define continuity of a function. (3 Marks)
ii) Show that the function ?? ?? =
??2-1 ??-1
,?? ? 1 2, ?? = 1 is continuous at x = 1 (3 Marks)
d) i) Show that the function ?? ?? =
2??2-3??+1 ??-1
,?? ? 1 5, ?? = 1
is discontinuous at x = 1.
(3 Marks) ii) Let ?? ?? = ??2-4??-5 ??-5 , for ?? ? 5. How should f(x) be defined so that f is continuous at ?? = 5? (3 Marks)
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